Theorem rmob 3135

Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 2-Jan-2015.) (Revised by NM, 16-Jun-2017.)

Hypotheses

Ref	Expression
rmoi.b	⊢ (x = B → (φ ↔ ψ))
rmoi.c	⊢ (x = C → (φ ↔ χ))

Assertion

Ref	Expression
rmob	⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ)) → (B = C ↔ (C ∈ A ∧ χ)))

Distinct variable groups:   x,A   x,B   x,C   ψ,x   χ,x

Allowed substitution hint:   φ(x)

Proof of Theorem rmob

Step	Hyp	Ref	Expression
1		df-rmo 2623	. 2 ⊢ (∃*x ∈ A φ ↔ ∃*x(x ∈ A ∧ φ))
2		simprl 732	. . . 4 ⊢ ((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → B ∈ A)
3		eleq1 2413	. . . 4 ⊢ (B = C → (B ∈ A ↔ C ∈ A))
4	2, 3	syl5ibcom 211	. . 3 ⊢ ((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → (B = C → C ∈ A))
5		simpl 443	. . . 4 ⊢ ((C ∈ A ∧ χ) → C ∈ A)
6	5	a1i 10	. . 3 ⊢ ((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → ((C ∈ A ∧ χ) → C ∈ A))
7		simplrl 736	. . . . 5 ⊢ (((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) ∧ C ∈ A) → B ∈ A)
8		simpr 447	. . . . 5 ⊢ (((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) ∧ C ∈ A) → C ∈ A)
9		simpll 730	. . . . 5 ⊢ (((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) ∧ C ∈ A) → ∃*x(x ∈ A ∧ φ))
10		simplrr 737	. . . . 5 ⊢ (((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) ∧ C ∈ A) → ψ)
11		eleq1 2413	. . . . . . 7 ⊢ (x = B → (x ∈ A ↔ B ∈ A))
12		rmoi.b	. . . . . . 7 ⊢ (x = B → (φ ↔ ψ))
13	11, 12	anbi12d 691	. . . . . 6 ⊢ (x = B → ((x ∈ A ∧ φ) ↔ (B ∈ A ∧ ψ)))
14		eleq1 2413	. . . . . . 7 ⊢ (x = C → (x ∈ A ↔ C ∈ A))
15		rmoi.c	. . . . . . 7 ⊢ (x = C → (φ ↔ χ))
16	14, 15	anbi12d 691	. . . . . 6 ⊢ (x = C → ((x ∈ A ∧ φ) ↔ (C ∈ A ∧ χ)))
17	13, 16	mob 3019	. . . . 5 ⊢ (((B ∈ A ∧ C ∈ A) ∧ ∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → (B = C ↔ (C ∈ A ∧ χ)))
18	7, 8, 9, 7, 10, 17	syl212anc 1192	. . . 4 ⊢ (((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) ∧ C ∈ A) → (B = C ↔ (C ∈ A ∧ χ)))
19	18	ex 423	. . 3 ⊢ ((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → (C ∈ A → (B = C ↔ (C ∈ A ∧ χ))))
20	4, 6, 19	pm5.21ndd 343	. 2 ⊢ ((∃*x(x ∈ A ∧ φ) ∧ (B ∈ A ∧ ψ)) → (B = C ↔ (C ∈ A ∧ χ)))
21	1, 20	sylanb 458	1 ⊢ ((∃*x ∈ A φ ∧ (B ∈ A ∧ ψ)) → (B = C ↔ (C ∈ A ∧ χ)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃*wmo 2205  ∃*wrmo 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-rmo 2623  df-v 2862

This theorem is referenced by:  rmoi  3136